Integrand size = 35, antiderivative size = 1105 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{x^4 \left (e+f x^2\right )^2} \, dx =\text {Too large to display} \] Output:
1/6*b*(-15*a*c*f+11*a*d*e+2*b*c*e)*x*(d*x^2+c)^(1/2)/a/e^3/(b*x^2+a)^(1/2) +1/6*(4*b^2*c*e*(3*c^2*f^2-5*c*d*e*f+2*d^2*e^2)-a^2*d*f*(30*c^2*f^2-37*c*d *e*f+10*d^2*e^2)+5*a*b*(-3*c^3*f^3+10*c^2*d*e*f^2-9*c*d^2*e^2*f+2*d^3*e^3) )*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a/c/e^3/(-a*f+b*e)/(-c*f+d*e)+1/6*d*(2 *b^2*c*e*(12*c^2*f^2-17*c*d*e*f+5*d^2*e^2)-a^2*d*f*(15*c^2*f^2-16*c*d*e*f+ 4*d^2*e^2)+a*b*(-30*c^3*f^3+49*c^2*d*e*f^2-26*c*d^2*e^2*f+4*d^3*e^3))*x^3* (b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a/c^2/e^3/(-a*f+b*e)/(-c*f+d*e)+1/6*b*d^2* (4*b*e*(3*c^2*f^2-4*c*d*e*f+d^2*e^2)-a*f*(15*c^2*f^2-16*c*d*e*f+4*d^2*e^2) )*x^5*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a/c^2/e^3/(-a*f+b*e)/(-c*f+d*e)-1/3* (b*x^2+a)^(3/2)*(d*x^2+c)^(5/2)/a/c/e/x^3/(f*x^2+e)-1/3*(-5*c*f+2*d*e)*(b* x^2+a)^(3/2)*(d*x^2+c)^(5/2)/a/c^2/e^2/x/(f*x^2+e)-1/6*f*(4*b*e*(3*c^2*f^2 -4*c*d*e*f+d^2*e^2)-a*f*(15*c^2*f^2-16*c*d*e*f+4*d^2*e^2))*x*(b*x^2+a)^(3/ 2)*(d*x^2+c)^(5/2)/a/c^2/e^3/(-a*f+b*e)/(-c*f+d*e)/(f*x^2+e)-1/6*b^(1/2)*( -15*a*c*f+11*a*d*e+2*b*c*e)*(d*x^2+c)^(1/2)*EllipticE(b^(1/2)*x/a^(1/2)/(1 +b*x^2/a)^(1/2),(1-a*d/b/c)^(1/2))/a^(1/2)/e^3/(b*x^2+a)^(1/2)/(a*(d*x^2+c )/c/(b*x^2+a))^(1/2)+1/6*a^(1/2)*b^(1/2)*(2*b*c*e*(-6*c*f+5*d*e)+a*(15*c^2 *f^2-16*c*d*e*f+3*d^2*e^2))*(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2) *x/a^(1/2)),(1-a*d/b/c)^(1/2))/c/e^3/(-a*f+b*e)/(b*x^2+a)^(1/2)/(a*(d*x^2+ c)/c/(b*x^2+a))^(1/2)-1/2*a^(3/2)*(-c*f+d*e)*(a*f*(-5*c*f+2*d*e)-b*e*(-4*c *f+d*e))*(d*x^2+c)^(1/2)*EllipticPi(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2)...
Result contains complex when optimal does not.
Time = 6.41 (sec) , antiderivative size = 443, normalized size of antiderivative = 0.40 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{x^4 \left (e+f x^2\right )^2} \, dx=\frac {\sqrt {\frac {b}{a}} \left (-\sqrt {\frac {b}{a}} e f \left (a+b x^2\right ) \left (c+d x^2\right ) \left (2 b c e x^2 \left (e+f x^2\right )+a d e x^2 \left (8 e+11 f x^2\right )+a c \left (2 e^2-10 e f x^2-15 f^2 x^4\right )\right )+i b c e f (-2 b c e-11 a d e+15 a c f) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i b e \left (-2 b c^2 e f+a \left (3 d^2 e^2-16 c d e f+15 c^2 f^2\right )\right ) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )-3 i a (-d e+c f) (b e (d e-4 c f)+a f (-2 d e+5 c f)) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right ) \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{6 b e^4 f x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \] Input:
Integrate[(Sqrt[a + b*x^2]*(c + d*x^2)^(3/2))/(x^4*(e + f*x^2)^2),x]
Output:
(Sqrt[b/a]*(-(Sqrt[b/a]*e*f*(a + b*x^2)*(c + d*x^2)*(2*b*c*e*x^2*(e + f*x^ 2) + a*d*e*x^2*(8*e + 11*f*x^2) + a*c*(2*e^2 - 10*e*f*x^2 - 15*f^2*x^4))) + I*b*c*e*f*(-2*b*c*e - 11*a*d*e + 15*a*c*f)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[ 1 + (d*x^2)/c]*(e + f*x^2)*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*b*e*(-2*b*c^2*e*f + a*(3*d^2*e^2 - 16*c*d*e*f + 15*c^2*f^2))*x^3*Sqrt[ 1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*(e + f*x^2)*EllipticF[I*ArcSinh[Sqrt[b/ a]*x], (a*d)/(b*c)] - (3*I)*a*(-(d*e) + c*f)*(b*e*(d*e - 4*c*f) + a*f*(-2* d*e + 5*c*f))*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*(e + f*x^2)*Elli pticPi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(6*b*e^4*f*x^3* Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2))
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{x^4 \left (e+f x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 450 |
| \(\displaystyle \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{x^4 \left (e+f x^2\right )^2}dx\) |
Input:
Int[(Sqrt[a + b*x^2]*(c + d*x^2)^(3/2))/(x^4*(e + f*x^2)^2),x]
Output:
$Aborted
Int[((g_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_)^2)^(p_.)*((c_.) + (d_.)*(x_)^2)^ (q_.)*((e_.) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Unintegrable[(g*x)^m*(a + b*x^2)^p*(c + d*x^2)^q*(e + f*x^2)^r, x] /; FreeQ[{a, b, c, d, e, f, g, m, p, q, r}, x]
Time = 21.02 (sec) , antiderivative size = 1458, normalized size of antiderivative = 1.32
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1458\) |
| risch | \(\text {Expression too large to display}\) | \(1468\) |
| default | \(\text {Expression too large to display}\) | \(2343\) |
Input:
int((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/x^4/(f*x^2+e)^2,x,method=_RETURNVERBOS E)
Output:
((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(-1/3*c/e^2*(b *d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/x^3+1/3/a*(6*a*c*f-4*a*d*e-b*c*e)/e^3*(b *d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/x+1/2*(c*f-d*e)*f/e^3*x*(b*d*x^4+a*d*x^2 +b*c*x^2+a*c)^(1/2)/(f*x^2+e)-8/3/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/ c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+ (a*d+b*c)/c/b)^(1/2))*b*c*d/e^2+1/2/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^ 2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(- 1+(a*d+b*c)/c/b)^(1/2))*b*d^2/e/f+11/6*c/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1 +d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*d*b/e^2*EllipticE(x*(- b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))+5/2/e^4*f^2/(-b/a)^(1/2)*(1+b*x^2/a)^ (1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticPi(x*( -b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*a*c^2-2/e^3*f/(-b/a)^(1/2 )*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)* EllipticPi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*b*c^2+5/2/e ^2/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x ^2+a*c)^(1/2)*EllipticPi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2 ))*b*c*d-1/2/e/f/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4 +a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticPi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/ 2)/(-b/a)^(1/2))*b*d^2+5/2*c^2/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^ (1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*b/e^3*f*EllipticF(x*(-b/a)^(1...
Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{x^4 \left (e+f x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/x^4/(f*x^2+e)^2,x, algorithm="fr icas")
Output:
Timed out
Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{x^4 \left (e+f x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate((b*x**2+a)**(1/2)*(d*x**2+c)**(3/2)/x**4/(f*x**2+e)**2,x)
Output:
Timed out
\[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{x^4 \left (e+f x^2\right )^2} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {3}{2}}}{{\left (f x^{2} + e\right )}^{2} x^{4}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/x^4/(f*x^2+e)^2,x, algorithm="ma xima")
Output:
integrate(sqrt(b*x^2 + a)*(d*x^2 + c)^(3/2)/((f*x^2 + e)^2*x^4), x)
\[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{x^4 \left (e+f x^2\right )^2} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {3}{2}}}{{\left (f x^{2} + e\right )}^{2} x^{4}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/x^4/(f*x^2+e)^2,x, algorithm="gi ac")
Output:
integrate(sqrt(b*x^2 + a)*(d*x^2 + c)^(3/2)/((f*x^2 + e)^2*x^4), x)
Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{x^4 \left (e+f x^2\right )^2} \, dx=\int \frac {\sqrt {b\,x^2+a}\,{\left (d\,x^2+c\right )}^{3/2}}{x^4\,{\left (f\,x^2+e\right )}^2} \,d x \] Input:
int(((a + b*x^2)^(1/2)*(c + d*x^2)^(3/2))/(x^4*(e + f*x^2)^2),x)
Output:
int(((a + b*x^2)^(1/2)*(c + d*x^2)^(3/2))/(x^4*(e + f*x^2)^2), x)
\[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{x^4 \left (e+f x^2\right )^2} \, dx=\int \frac {\sqrt {b \,x^{2}+a}\, \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{x^{4} \left (f \,x^{2}+e \right )^{2}}d x \] Input:
int((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/x^4/(f*x^2+e)^2,x)
Output:
int((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/x^4/(f*x^2+e)^2,x)